
where \(\hbox{$\phi $}_F^0\)  is the (global) lowest-order basis for
the lowest-order Raviart-Thomas space whose normal component is only supported
on the facet \(F\) ,
\(\hbox{$\phi $}_F^i\) ,
\(i=1,\cdots , n_f^r\) ,
is the
(global) higher order divergence-free facet bubble basis whose normal component
is only supported on the facet \(F\) ,
\(\hbox{$\phi $}_K^i\) ,
\(i=1,\cdots , n_k^{r,1}\) ,
is the (local)
higher order divergence-free element bubble
basis
that is supported only on the element \(K\) 
whose normal component vanishes on the mesh skeleton \(\mathcal {E}_h\)  and
\(\hbox{$\psi $}_K^i\) , \(i=1,\cdots , n_k^{r,2}\) , is the (local)
higher order element bubble basis
with a non-zero divergence
that is supported only on the element \(K\) 
whose normal component vanishes on the mesh skeleton \(\mathcal {E}_h\) .
Here the integer numbers \(n_f^r\) , \(n_k^{r,1}\) ,
and \(n_k^{r,2}\)  are the number of basis functions per facet/element for each group
of basis functions, whose specific values
can be found in [1]} which are not relevant in our discussion.
With this basis splitting, we now define the global and local
subspaces of \(\hbox{$V$}_h^r\) :
\(\hbox{$V$}_h^{r,\partial } := &\;\bigoplus _{\tiny F\in \mathcal {E}_h} \mathrm {span}\lbrace \hbox{$\phi $}_F^0\rbrace \oplus \!\!\!\!\!\!\!\!\!\!\bigoplus _{\tiny \begin{tabular}{c}F\in \mathcal {E}_h\\i=1,\cdots , n_f^r\end{tabular}}\!\!\!\!\!\!\!\!\!\!\mathrm {span}\lbrace \hbox{$\phi $}_F^i\rbrace \\\hbox{$V$}_h^{r,o} :=&\;\!\!\!\!\!\!\!\!\!\!\bigoplus _{\tiny \begin{tabular}{c}K\in \mathcal {T}_h\\i=1,\cdots , n_k^{r,1}\end{tabular}}\!\!\!\!\!\!\!\!\!\!\mathrm {span}\lbrace \hbox{$\phi $}_K^{i}\rbrace ,\;\;\;\;\;\;\;\;\;\;\hbox{$V$}_h^{r,o*} :=\;\hbox{$V$}_h^{r,o}\oplus \!\!\!\!\!\!\!\!\!\!\bigoplus _{\tiny \begin{tabular}{c}K\in \mathcal {T}_h\\i=1,\cdots , n_k^{r,2}\end{tabular}}\!\!\!\!\!\!\!\!\!\!\mathrm {span}\lbrace \hbox{$\psi $}_K^{i}\rbrace .\) 
